So my son and I just killed the Dragon on the easy dungeon, do we each get to check off 9 green circles? We both finished 3 floors, leveled up to 4 and defeated the boss. But do we both get 9 greens or split them?

Then can I use that same character and play alone and add more and just keep using that character in one and two player games?

Each get the points. And transfering the character is up to you. I, personally, note if the character is 1p or 2p. But there was never any mention of localizing like that. It was more true campaign where the character is the same no matter the interation.